Unbelievably competitive. I know a person going to Oxford and multiple people going to Trinity College in Dublin (although no one for Cambridge) and they are all incredibly intelligent and accomplished. You don't just apply, they ask you to apply. That's not to say you won't get it, but it's no walk in the park.
Notre Dame does have an excellent program in London, which is fairly reasonable to get into provided you express the right amount of interest. Also, Notre Dame's program in Dublin, UCD (which I am attending next semester) is considered one of the best, and it is also very competitive (although not like Oxford).
There are tons of options for Study Abroad in the UK, not just the Oxford and Cambridge program. Tons of options elsewhere, as well. We have one of the highest percentages of students studying abroad among American universities. Check out
Office of International Studies - University of Notre Dame It will satisfy your curiosity for a while.